Condensed Matter, Marder, Solutions

Solutions to Problems in Condensed Matter Physics 2 (Textbook: Michael P. Marder, Condensed Matter Physics, Wiley, 2000)

Contents Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter

page

6 and 7 (“electrons”) 11 (“phonons”) 12 (“elasticity”) 17 (“transport phenomena”) 22 and 23 (“optical properties”) 24 (“ordering”) 25 and 26 (“magnetism”) 27 (“superconductivity”)

1 6 8 10 16 21 26 33

(Niels Bohr Institute, February 2011)

1

Problems in Chapter 6 and 7 (“electrons”)

Solutions to the problems in Chapter 6 6.3 Pressure of a Fermi gas at zero temperature The number of electrons N and the internal energy U , in the volume V , are ∞

N =V

∞

D(ε)f (ε)dε,

0

U =V

0

εD(ε)f (ε)dε,

(1)

The Fermi distribution function f (ε) and the density of states (per unit volume) D(ε) are √ √ 1 2m3 (2) , D(ε) = A ε, A = 2 3 . f (ε) = β(ε−µ) π e +1 At zero temperature 1/3 2 kF2 , kF = 3π 2 n (3) 2m √ Introducing the step function and D(ε) = A ε in (1), we may determine U (0) and an alternative expression for A: ε √ F 3n 3/2 5/2 A εdε = 23 V AεF ⇒ A = 3/2 , U (0) = V 25 AεF = 35 N εF (4) N =V 0 2εF

f (ε) = θ(εF − ε),

εF =

i.e. U ∝ εF ∝ kF2 ∝ V −2/3 and the pressure at zero temperature is (3π 2 )2/3 n5/3 2 ∂U (0) 2U = = 25 nεF = (5) ∂V 3V 5m in sharp contrast to a classical gas, where the zero-temperature pressure is zero. P =−

6.4 Density of states in low dimensions The density of states in k-space is Dk = 2/(2π)d in d dimensions (6.34)

D(ε) =

[dk]δ(ε − εk ) ≡

Dk δ(ε − εk )dk =

2 (2π)d

δ(ε − εk )dk

(1)

Introducing “spherical coordinates” in d dimensions and k = |k|, then dk =

⎧ ⎨ 2 dk

2πk dk ⎩ 4πk2 dk

, d=1 , d=2 , d=3

(2)

The factor 2 in the one-dimensional case appears because the one-dimensional wave vector may assume both positive and negative values, whereas k ≥ 0 per definition. 2 k2 εk = εk = 2m

D(ε) =

⇒

k=

2mεk ,

dk =

⎧ 1/2 ∞ ⎪ 2 m ⎪ ⎪ ⎪ 2 δ(ε − εk ) dεk = ⎪ ⎪ 2π 0 22 εk ⎪ ⎪ ⎪ 1/2 ⎪ ∞ ⎨

2mεk

2

m 22 εk

0

dεk

2 εk

⇒

(3)

√

2m −1/2 , d=1 ε π m , π2

d=2

2m3 1/2 ε , π 2 3

d=3

m

2π dεk = δ(ε − εk ) ⎪ (2π)2 0 22 εk ⎪ ⎪ ⎪ 1/2 ⎪ ⎪ ∞ ⎪ ⎪ 2mε m 2 ⎪ ⎪ 4π 2 k δ(ε − εk ) dεk = ⎩ 3 2 (2π)

1/2

√

Condensed Matter Physics 2

2

6.5 Fermi pancakes A and Lz = d. Thin layer of Ag: Lx = Ly = L = 106 ˚ The density of electrons is the same as the density of atoms (one conduction A−3 , according to the electron per Ag atom), i.e. n = 5.86 · 1022 cm−3 = 0.0586 ˚ properties given in the periodic table on the front page of Marder. The wave function of the “free” electrons should vanish at the boundaries z = 0 and z = d. This condition is fulfilled if assuming the one-electron wave function to be ψ(x, y, z) ∝ ei(kx x+ky y) sin (pqz) ,

q=

π , d

p = 1, 2, . . .

(1)

The electron states are characterized by the two-dimensional wave vector k = (kx , ky , 0) and the “band index” p. The eigenenergies are (k2 = kx2 + ky2 ): εpk =

2 2 2 q 2 k + p2 q 2 ≥ ε10 = 2m 2m

(2)

At T = 0 (T TF ), as assumed implicitly in the exercise, the occupied states are all those with energies smaller than εF . The number of electrons in the pth band, Np , is zero if εp0 > εF . In the opposite case: Np = L2

k p 0

2 2 2 kp 2πk dk = L (2π)2 2π

(3)

where kp is the Fermi wave number of the pth band, i.e. the largest value of k of occupied states in the pth band, as determined by εpk

p

2 2 = kp + p2 q 2 = εF 2m

kp2

⇒

2mεF = − p2 q 2 = q 2 2

εF − p2 ε10

(4)

˚−1 , and (a) In the first case d = 4.1 ˚ A or q = π/d = 0.76624 A ε10 =

(1.054572 · 10−27 × 0.76624 · 108 )2 2 q 2 = eV = 2.2369 eV 2m 2 × 9.109389 · 10−28 × 1.602177 · 10−12

(5)

Assuming the Fermi energy to be smaller than the lowest energy of the (p = 2) band, εF < ε20 = 4ε10 , then only the (p = 1) states are occupied, in which case N1 has to be equal the total number N of free electrons: N = L2 d n = N1 = L2

k12 2π

⇒

k1 =

√

−1 2πd n = 1.22866 ˚ A

(6)

Introducing this result in (3) we get the Fermi energy and the band width W of the occupied states:

εF = ε1k = ε10 (k1 /q)2 + 1 = 7.99 eV, 1

W = εF − ε10 = 5.75 eV

(7)

which is in accordance with our starting assumption of εF < ε20 = 8.948 eV. These results may be compared with that obtained for bulk Ag: εF (bulk) = W (bulk) =

2 2 2/3 3π n = 5.50 eV 2m

(8)

3


(b) In the case of d = 8.2 ˚ A, q = π/d = 0.38312 ˚ A−1 and ε10 = 0.5592 eV. In order to determine εF in this case, we need to include more bands than the lowest one. The number of bands turns out to be 3, and

N = N1 + N2 + N3 ⇒ 2πd n = q

2

εF − 12 + ε10

εF − 22 + ε10

εF − 32 ε10

(9)

Solving this equation with respect to εF /ε10 , we get εF /ε10 = 11.523, which is larger than 32 but smaller than 42 , in accordance with the assumption that all electrons are found in the three lowest bands. Hence the results are: W = εF − ε10 = 5.89 eV

εF = 6.44 eV, 20

(10)

p 2

Band energy (eV)

16 12

1

εF

8 4

k1

0 0.0

0.2

0.4 0.6 0.8 Wave number k [r.l.u.]

1.0

5

20

p 4

Band energy (eV)

16 3

12

2 1

8

εF

4

k3

0 0.0

0.2

k2

k1

0.4 0.6 0.8 Wave number k [r.l.u]

1.0

Fermi energy and band width (eV)

12

εF

10 8 6

W

4

Bulk value

2 0

0

4 8 12 16 Number of Ag layers (2d/a)

20

The two first figures show the energy bands as functions of k in the cases (a) and (b), where the unit of k is the length of a reciprocal lattice vector 2π/a [r.l.u.]. The last figure shows the Fermi energy (and the band width) as a function of the number of atomic layers of Ag. The crystal structure of Ag is fcc with the lattice parameter a = 4.09 ˚ A. This is very nearly the thickness d assumed in (a), i.e. this case corresponds to a film with two atomic layers of Ag atoms.


4

Solutions to the problems in Chapter 7 7.1 Normals to surfaces r = (x1 , x2 , x3 ) = s(t) is the parametrization of a curve lying within the surface defined by f (r ) = ε. Since f (s(t)) is a constant ε, the derivative of this function is 0: ∂f dsα ds d f (s(t)) = = ∇f · =0 (1) dt ∂x dt dt α α Because s(t) may be any arbitrary curve lying within the surface, the same is true for the curve tangent ds(t)/dt, and (1) is only generally valid if ∇f is normal to the surface. 7.3 Van Hove singularities (a) The problem becomes the same as the one considered in problem 7.1 if making the replacements k → r and ε → f (r), hence ∇ ε is perpendicular to the k nk nk energy surface defined by εnk = ε. (b) The energy is assumed to be εnk = εmax − k2 , and in the two-dimensional case

D(ε) =

[dk]δ(ε − εnk ) =

1 = 2π

∞ 0

2 (2π)2

∞ 0

2πk δ(ε − εmax + k2 ) dk (1)

1 θ(ε δ(ε − εmax + k )d(k ) = − ε) 2π max 2

2

The density of states is zero if ε > εmax and 1/(2π) when ε < εmax . (c) In the three dimensional case, the result is

[dk]δ(ε − εnk ) =

D(ε) = =

1 2π 2

∞ 0

2 (2π)3

∞ 0

4πk2 δ(ε − εmax + k2 ) dk

k δ(ε − εmax + k2 )d(k2 ) =

1 εmax − ε θ(εmax − ε) 2π 2

(2)

Solution to HS’s problem 1 Heat capacity of a two-dimensional electron gas GaAs/AlGaAs heterostructure (see Marder Section 19.5): n = 1011 cm−2 ,

ε=

2 k2 , 2m∗

m∗ = 0.067me

(1)

The present situation corresponds to the case (a) of the previous problem 6.5 in Marder, i.e. the gas is purely two-dimensional in the sense that only the (p = 1) band needs to be considered, and k is a two-dimensional vector with the length k = 2 kx + ky2 . The most important quantity is the Fermi energy, which is determined by evaluating N at zero temperature: n=

N = A

k≤kF

Dk dk =

2 (2π)2

k F 0

2πk dk =

kF2 2π

⇒

kF =

√

2πn

(2a)

5


The same result is obtained by using that, according to problem 6.4 or equation (6.35), D(ε) = m∗ /π2 in the two-dimensional case: n=

ε F 0

D(ε) dε =

m∗ εF π2

⇒

εF =

nπ2 m∗ −1

A Introducing the numbers, and using that kF = 1 ˚ εF = 3.81 eV when the mass is me , then we get kF = 0.793 · 10

−2

−1 ˚ A ,

or

kF =

√

2πn

(2b)

corresponds to a Fermi energy

2

0.793 · 10−2 εF = 3.81 eV = 3.58 meV 0.067

(3)

This Fermi energy corresponds to a Fermi temperature TF = εF /kB = 41.5 K. 1) T = 1 K is much smaller than the Fermi temperature and the heat capacity may be determined by the leading order expression (6.77) cV =

π2 π T 2 2 D(εF )kB T = k k , 3 6 TF F B

as

D(εF ) =

m∗ kF2 kF2 = = (4) π2 2πεF 2πkB TF

In the case of a sample with the area A = 1 cm2 the result is π 1 (0.793·106 cm−1 )2 ×1.38066·10−23 J/K = 1.1 · 10−13 J/K 6 41.5 (5) ∗ 2 2 The explicit result is CV (el) = Aπm kB T /(3 ). Hence, the small value of CV is not due to the low electron density n but to the two-dimensionality of the system and the small effective mass. CV (el) = A cV = A

2) In order to estimate the phonon contribution to the heat capacity we shall use the Debye model (Section 13.3.2 in Marder). The Debye temperature of GaAs is ΘD = 344 K (which value is not changed much when some of the Ga ions are replaced by Al ions). Using (13.70), (13.75) in Marder (2.Ed.) and ∞ 0

4π 4 x4 ex dx = (ex − 1)2 15

⇒

12π 4 cV = n k 5 A B

T ΘD

3

(T ΘD )

(6)

According to Table 2.5 (page 27) in Marder, GaAs has the zincblende structure (∼ diamond structure) with the lattice parameter a = 5.63 ˚ A. In this structure there are 8 atoms per unit cell, and the density of atoms is nA = 8/a3 = 4.48 · 1022 cm−3 . [The mean atomic mass (periodic table) is (69.72 + 74.92)/2 = 72.3 u, implying a mass density ρ = 5.38 g cm−3 ]. Assuming V = 1 cm3 and T = 1 K, the phonon contribution becomes

CV (ph) =

12π 4 1 × 4.48 · 1022 × 1.38066 · 10−23 5 344

3

J/K = 3.55 · 10−6 J/K (7)

which is much larger than the electronic contribution. Utilizing that CV (el) ∝ T and CV (ph) ∝ T 3 , the temperature T0 at which the two contributions are equal, is determined by (T0 in K) 1.10 · 10−13 T0 = 3.55 · 10−6 T03

⇒

T0 = 1.8 · 10−4 K

(8)


6

A more fair comparison would be to consider a film of thickness ∼ 100 µm, i.e. A = 1 cm2 and V = 10−2 cm3 , in which case T0 = 1.8 mK, a temperature within an accessible range (however, the reduction of the size of the sample makes it more difficult to determine the heat capacity).

Solution to the problem in Chapter 11 11.2 Zinc in copper Copper is a monovalent fcc and zinc a divalent hcp (not fcc) metal. At a small concentration of Zn++ , the crystal structure is fcc until the Fermi surface touches some point on the edge of the Brillouin zone: (a) In the nearly free electron approximation, the Fermi wave vector is (nearly) determined as in the free electron case,

n=

|k|≤kF

Dk dk =

2 (2π)3

k F 0

4πk2 dk =

1 3 k 3π 2 F

kF = 3π 2 n

⇒

1/3

(1)

Defining c to be the concentration of Zn++ ions and a to be the lattice parameter of the fcc lattice, then the electron density is 4 4 n = 3 [(1 − c) + 2c] = 3 (1 + c) a a

1/3

12π 2 kF = (1 + c) a3

4.9109 (1 + c)1/3 a (2) (b) The primitive unit vectors of the face-centered cubic lattice are, (2.2): a1 =

a (1, 1, 0), 2

⇒

a2 =

a (1, 0, 1), 2

a3 =

=

a (0, 1, 1) 2

(3)

and the corresponding primitive vectors of the reciprocal lattice are b = 2π a2 × a3 = 2π (1, 1, −1), 1 a1 · a2 × a3 a

b = 2π (1, −1, 1), 2 a

b = 2π (−1, 1, 1) 3 a

(4) The boundaries of the 1. Brillouin zone are established by the planes perpendicular to ±bi at the distance k1 = |bi |/2 from the origin. Hence the shortest distance from the origin to the zone boundaries is √ π 3 5.4414 |bi | = = k1 = 2 a a

(5)

kF (c = 0) is smaller than k1 , and kF (c) becomes equal to k1 at the zinc concentration c = c1 4.9109(1 + c1 )1/3 = 5.4414 ⇒ c1 = 0.36 (6) The experimental phase diagram (see Fig. 4 in Chapter 21 of Kittel) shows a transition from fcc to bcc, for increasing c values, at about the Zn concentration determined here.

7

Problems in Chapter 11 (“phonons”)

(c) The primitive unit vectors of the body-centered cubic lattice are, (2.3): a1 =

a (1, 1, −1), 2

a2 =

a (1, −1, 1), 2

a3 =

a (−1, 1, 1) 2

(7)

and the corresponding primitive vectors of the reciprocal lattice are b = 2π (1, 1, 0), 1 a

b = 2π (1, 0, 1), 2 a

b = 2π (0, 1, 1) 3 a

(8)

Notice, that the reciprocal of the bcc lattice is fcc, and equivalently, that the reciprocal lattice of fcc is the bcc lattice. In the case of bcc there are 2 atoms per cubic unit cell, hence 2 n = 3 (1 + c), a

1/3

6π 2 kF = (1 + c) a3

=

3.8979 (1 + c)1/3 a

(9)

The shortest distance from the origin to the boundaries of the 1. Brillouin zone is √ π 2 4.4429 |bi | = = k2 = 2 a a

and k2 = kF

when

c = c2 = 0.48

(10)

Once again, this is about the right Zn concentration at which the Cu-Zn alloy system shows a change of crystal structure (from bcc to a complex γ structure). If the atomic density is assumed to stay constant at the transition fcc → bcc, the Fermi wave vector is the same on each side of the phase line, but the lattice parameter a is changed, abcc = (1/2)1/3 afcc = 0.794 afcc . The cubic lattice parameter a also changes (increases) gradually with the Zn concentration, in between the phase lines, since nCu = 1.33 nZn , however, all the possible variations of a have no influence on the arguments above. β-brass is the alloy with c = 0.5, and it shows an order-disordered phase transition at about 470 ◦ C, which we are going to discuss later on. Citation from Kittel: “Why is there a connection between the electron concentrations at which a new phase appears and at which the Fermi surface makes contact with the Brillouin zone? We recall that the energy bands split into two at the region of contact on the zone boundary [Marder, Chapter 8, (8.24)]. If we add more electrons to the alloy at this stage, they will have to be accommodated in the upper band or in states of high energy near the zone corners of the lower band. Both options are possible, and both involve an increase of energy. Therefore it may be energetically favorable for the crystal structure to change to one which can contain a Fermi surface of larger volume (more electrons) before contact is made with the zone boundary. In this way H. Jones made plausible the sequence of structures fcc, bcc, γ, hcp with increasing electron concentration.”


8

Solutions to the problems in Chapter 12 12.4 Elastic constants (a) The application of a uniform external gas pressure (no shear stress) implies a uniform dilations of an isotropic solid, where only exx = eyy = ezz are non-zero, or eαβ = δαβ

∆V V − V0 = 3V0 3V0

(1)

where V0 is the zero-pressure equilibrium volume. Introducing this in the free energy expression for an isotropic solid (12.23)

F=

1 2

λ

eαα

2

+ 2µ

α

e2αβ

dr =

αβ

1 2 (V0

∆V + ∆V ) λ V0

2

3 ∆V + 2µ 9 V0

2

(2) the bulk modulus is found to be B=V

∂2F = λ + 23 µ ∂V 2

(3)

(when V → V0 ). This result is also obtained from the bulk modulus in the cubic case (12.19), B = [c11 + 2c12 ]/3, by replacing the elastic constants with the Lamé constants of the isotropic solid, c12 = λ, c44 = µ, and c11 = c12 + 2c44 = λ + 2µ. (b) According to (12.31), or Fig. 12.2, Young’s modulus Y is defined to be Y = σzz /ezz , when applying a uniform stress σzz in the z direction. In the cubic case, the diagonal elements of stress and strain tensors are related by, (12.14)-(12.17), ⎛

⎞

⎛

⎞

σxx exx ⎝σ ⎠ = c⎝e ⎠, yy yy σzz ezz

⎛

c11 c = ⎝ c12 c12

c12 c11 c12

⎞

c12 c12 ⎠ c11

(4)

[the “off-diagonal elements relations” are σα = c44 eα , where α = 4, 5, 6. In the case of α = 4 this equation reads σyz = 2c44 eyz ]. In order to determine ezz when σzz is non-zero, we need to determine the inverse matrix. The determinant of c is D = c311 + 2c312 − 3c11 c212 = (c11 + 2c12 )(c11 − c12 )2 , and the inverse matrix is found to be ⎛

c

−1

A = ⎝B B

B A B

⎞

B B ⎠, A

A=

c11 + c12 , (c11 + 2c12 )(c11 − c12 )

B=−

c12 A c11 + c12

(5)

showing that ezz = Aσzz , hence Y = A−1 =

(c11 + 2c12 )(c11 − c12 ) c11 + c12

(6)

[In order to complete the cubic case, then Poisson’s ratio (12.34) is −B/A or c12 , and the shear modulus is G = c44 ]. ν= c11 + c12

9

Problems in Chapter 12 (“elasticity”)

12.5 Waves in cubic crystals (a) Combining (12.14) and (12.16) we may write the free energy of a cubic solid as

F=

1 2

dr c11 e2xx + e2yy + e2zz + 2 c12 exx eyy + exx ezz + eyy ezz

+2 c44

e2xy

+

e2yx

+

e2xz

+

e2zx

+

e2yz

e2zy

+

=

1 2

dr

(1) σαβ eαβ

αβ

which shows that the stress-strain relations are σαα = c11 eαα + c12

β=α

eββ ,

σαβ = 2 c44 eαβ (α = β),

eαβ

1 ≡ 2

∂uβ ∂uα + ∂rβ ∂rα

(2) Introducing these relations in the equations of motion, (12.25), we get ρ¨ uα =

∂σαβ

∂rβ

β

∂ 2 uβ ∂ 2 uα ∂ ∂uα ∂uβ = c11 + c12 + c44 + ∂rα2 ∂rα ∂rβ ∂rβ ∂rβ ∂rα β=α

(3)

(b) Introducing a plane-wave solution

u(r , t) = u0 ei(k·r−ωt)

(4)

into these equations (3), we get −ρω 2 u0α = −c11 kα2 u0α − c12

kα kβ u0β − c44

β=α

β=α

kβ2 u0α + kα kβ u0β

(5)

leading to the following matrix equation for determining u0 and ω

c11 kα2

+ c44

β=α

kβ2

− ρω

2

u0α + (c12 + c44 )

β=α

kα kβ u0β = 0

(6)

(c) In the case of waves propagation along [100], the k-vector is k = (k, 0, 0) and the matrix equation is diagonal:

c11 k2 − ρω 2 u01 = 0,

c44 k2 − ρω 2 u02 = 0,

c44 k2 − ρω 2 u03 = 0

(7)

The sound velocities are

ω c11 = , k ρ ω c44 , ct = = k ρ

cl =

u0 k

(longitudinal)

u0 ⊥ k

(transverse)

(8)

i.e. one longitudinal and two degenerate transverse sound waves. Table 12.1 shows that c11 = 165 GPa, c44 = 79.24 GPa, and c12 = 64 GPa in the case of silicon. From the periodic table: the atomic density is n = 4.99 · 1022 cm−3 and the atomic mass is 28.09 u implying ρ = 28.09 × 1.66054 · 10−27 [kg] × 4.99 · 1028 [m−3 ] = 2328 kg/m3 . Introducing these numbers in (8), we get cl = 8420 m/s ,

ct = 5830 m/s

(9)


10 √

(d) Waves propagating along [111], in which case we assume k = (k/ 3)(1, 1, 1), and the eigenvalue equation for determining λ = ρω 2 is found to be ⎛

⎞⎛

⎞

⎛ ⎞

A−λ B B u01 0 ⎝ B A−λ B ⎠ ⎝ u02 ⎠ = ⎝ 0 ⎠ u03 B B A−λ 0 where

λ = ρω 2 ,

A = 13 (c11 + 2c44 )k2 ,

The characteristic determinant is

B = 13 (c12 + c44 )k2

(10a)

(10b)

(A−λ) (A − λ)2 − B 2 −2B (A − λ) − B 2 = (A+2B−λ)(A−B−λ)2 = 0 (11) Introducing the solution λ = A + 2B in (10a), we get u01 = u02 = u03 or u0 k, whereas the two other degenerate solutions λ = A − B imply u01 + u02 + u03 = 0 or u0 · k = 0. Hence the sound velocities are

ω c11 + 2c12 + 4c44 = 9340 m/s , cl = = k 3ρ ω c11 − c12 + c44 = 5080 m/s , ct = = k 3ρ

u0 k

(longitudinal) (12)

u0 ⊥ k

(transverse)

Solutions to the problems in Chapter 17 17.4 AC conductivity In the presence of a time-dependent (uniform) electrical field =E e−iωt E 0

(1)

we may use the general solution of the Boltzmann equation given by (17.24), before the integration with respect to t is performed: g=f−

t

0 e ∂f (t ) dt e(t −t)/τε e−iωt vk · E ∂µ −∞

(2)

Considering only effects which are linear in the applied field (the linearized solution of the Boltzmann equation), then vk = vk (t ) and f (t ) within the integral may be replaced by their time-independent equilibrium values at zero field, and the time integration may be performed straightforwardly g=f−

τε τε e ∂f e−iωt = f − e ∂f vk · E vk · E 0 1 − iωτε ∂µ 1 − iωτε ∂µ

(3)

Introducing this expression into (17.43)-(17.44) we get the frequency-dependent → 0) conductivity (valid in the limit of E 0 σαβ (ω) = e2

[dk]

τε ∂f vα vβ 1 − iωτε ∂µ

(4)

or in the case of a cubic or an isotropic (free electron) system: σ(ω) =

ne2 ne2 τ 1 + iωτ τ = m∗ 1 − iωτ m∗ 1 + (ωτ )2

(5)

11

Problems in Chapter 17 (“transport phenomena”)

17.5 Current driven by thermal gradient We shall consider a metal subject to a constant temperature gradient

∇T =

∂T , 0, 0 ∂x

εk = εk = 12 m∗vk2

and

(1)

According to (17.60), (17.62), and (17.68) the electrical current, in the case of G = 0, is

j = L12 − ∇T T

,

1 1 π2 L12 = − L(1) = − (k T )2 σ (εF ) e e 3 B

(2)

The assumption of an isotropic mass, (1), implies σ(ε) to be diagonal, and according to (17.64) the diagonal element is 2

σαα (ε) = e τ

2 dkDk vkα δ(ε − εk ) = e2 τ

dεk D(εk ) 13 vk2 δ(ε − εk )

(3)

2 over all solid angles, at a constant | k|, is 1/3 of the result The integration of vkα 2 2 2 ∗ deriving from Tr v = v and using v = 2εk /m , we get kα

k

σαα (ε) = e2 τ D(ε)

k

2ε 3m∗

⇒

σαα (ε) =

2e2 τ D(ε) + ε D (ε) 3m∗

(4)

This result is introduced in (2)

2 2 j = 1 π (kB T )2 2e τ D(εF ) 1 + εF D (εF ) e 3 3m∗ D(εF )

∇T T

(5)

2 T D(ε ), (6.77), we finally get In terms of the heat capacity cV = (π 2 /3)kB F

2eτ cV ∂T jx = 3m∗ ∂x

D (εF ) 1 + εF D(εF )

where the last equality sign is valid only if D(ε) ∝

= √

eτ cV ∂T m∗ ∂x

(6)

ε.

17.8 Hall effect – elementary argument

is along the x axis leading to a current j in The Hall effect geometry: The applied field E this direction, i.e. an electron (hole) current in the minus (plus) x direction. The magnetic is in the minus y direction, when B is along z, leading part of the Lorenz force F ∝ j × B to opposite signs of the resulting charge distributions in the electron and hole cases.


12

(a) The equation of motion, when assuming the Drude model, is

− mv + v × B mv˙ = −e E c τ

(1)

= (0, 0, B), then we get v × B = in the case of electrons with charge −e. Using B (vy B, −vx B, 0) and since the current, by geometry, is constrained to be along x, the steady state is characterized by vx being constant and vy = 0. These conditions imply vx v mv˙ y = −e Ey − B − 0 = 0 ⇒ Ey = x B (2) c c (b) The current is j = (jx , 0, 0) with jx = −nevx , and the Hall coefficient is

R=

Ey 1 1 v = − = xB , B jx c B(−nevx ) nec

Ey = RB jx

(3)

The electric field in the x direction may be determined from v˙ x = 0, or mv˙ x = −eEx −

mvx =0 τ

⇒

Ex = −

me me jx mj j vx = − = 2x = x τ τ (−ne) ne τ σ

(4)

i.e. Ex is determined by the Drude resistivity σ −1 as in the case of B = 0. 17.9 Hall effect – Boltzmann equation The Boltzmann equation in the relaxation-time approximation is given by (17.17) and (17.18) dg g−f ∂g ˙ ∂g ˙ ∂g =− = + r · +k· (1) dt ∂t ∂r τ ∂k When the state is uniform in space and steady in time, this equation reduces to k˙ · ∂g = − g − f τ ∂k

(2)

The semiclassical equation of motion is

˙ + 1 v × B , k = −e E c k

1 ∂εk k vk = r˙ = = ∗ ∂k m

(3)

where the second equation expresses that the mass tensor is assumed to be isotropic for simplicity. Like in Problem 17.4 we are only interested in the response (current) which is linear in the electric field. This means that g may be replaced by f in The linearized products on the left hand side of (2) which already involve E. version of the Boltzmann equation (2) is therefore e e ∂f · ∂(g − f ) = − g − f · − vk × B − E c τ ∂k ∂k

(4)

· ∂f = 0 because ∂f is parallel with v , as when using that vk × B k ∂k ∂k ∂f ∂f ∂εk ∂f = = − vk ∂ε ∂µ ∂k k ∂ k

(5)

13


= (a) The geometry is the same as applied in problem 17.8, hence we define B (0, 0, B) and assume the resulting E = (Ex , Ey , 0) to be perpendicular to B. In this geometry, we guess that the solution has the form g = f + a kx + b ky

(6)

Introducing this in eq. (4) and utilizing (5), we get a kx + b ky eB e ∂f E k + E k − a k − b k =− x x y y y x ∗ ∗ m ∂µ m c τ

(7)

Since kx and ky are independent variables, this equation leads to two independent conditions, which determine a and b to be Ex − ωc τ Ey a= 1 + (ωc τ )2

eτ − ∗ m

∂f , ∂µ

Ey + ωc τ Ex b= 1 + (ωc τ )2

where we have introduced the cyclotron frequency ωc =

eτ − ∗ m

∂f ∂µ

(8)

eB . m∗ c

The α component of the current density j is, according to Marder’s eq. (17.43), jα = −e =−

[dk] vkα (gk − fk ) = −e

em∗

[dk] vkα a kx + b ky

2 2 [dk] a vkx δαx + b vky δαy

(9)

where the last equality sign follows because the off-diagonal terms vanish, when the mass tensor is assumed to be diagonal (isotropic). Using the same procedure as in Marder’s eqs. (17.44)-(17.50), we have −

em∗

eτ 2 [dk] vkα − ∗ m

∂f = e2 τ ∂µ

ne2 τ 2 ∂f = [dk] vkα ≡ σ0 ∂µ m∗

(10)

and combining the three equations (8)-(10), we finally get jx =

Ex − ωc τ Ey σ , 1 + (ωc τ )2 0

jy =

Ey + ωc τ Ex σ 1 + (ωc τ )2 0

(11)

is assumed to be along the z axis parallel to the field, we have In the case where E to add a term cz kz to the trial function g in (6). In this situation, the magnetic field does not contribute to the Boltzmann equation, and we get jz = σ0 Ez . Hence, for a system with an isotropic mass m∗ , the total conductivity tensor is found to be ⎛ ⎞ 1 −ωc τ 0 σ0 eB ⎝ω τ ⎠, (12) σ= 1 0 ωc = ∗ c 2 1 + (ωc τ ) m c 0 0 1 + (ω τ )2 c

when the magnetic field B is applied along the z axis. This result may also be expressed in terms of the Hall coefficient R, where ωc τ = −RBσ0 ,

or

R=−

1 nec

(13)


14

= ρ j, is the inverse of the The resistivity tensor ρ, defined by the relation E conductivity tensor and is particularly simple ⎛

ρ=σ

−1

1 ⎝ 1 = −ωc τ σ0 0

⎞

ωc τ 1 0

⎛

0 1 ⎝ 1 0⎠ = σ0 RB σ0 1 0

−σ0 RB 1 0

⎞

0 0⎠ 1

(14)

which is in perfect agreement with the results derived from the Drude model (problem 17.8). The calculations in section 17.4.8 assume ωc τ 1, in which case the diagonal elements of σ may be neglected in comparison with the off-diagonal ones, σxx = σyy 0 and σxy = −σyx −(ωc τ )−1 .

Solution to HS’s problem 3 Hall effect of a two-dimensional electron gas A two-dimensional electron gas with an anisotropic dispersion ε = a(kx2 + ky2 ) + b(kx4 + ky4 ),

a > 0,

b>0

(1)

Introducing the polar angle θ in the (kx , ky )-coordinate system of the reciprocal lattice, the dispersion relation may be written ε = a k2 + 14 b (3 + cos 4θ)k4 ,

kx = k cos θ,

ky = k sin θ

(2)

reflecting directly the four-fold, cubic symmetry of the dispersion. 1) The equation determining the constant energy contour is obtained by solving (2) with respect to k2 2a k = k (ε) = b(3 + cos 4θ) 2

2

1/2

bε 1 + 2 (3 + cos 4θ) a

−1

(3)

In a2 , the square root may be expanded, and to second order √ the case of bε ( 1 + x = 1 + 12 x − 18 x2 ) the result is

ε 3bε k (ε) 1− 2 a 4a 2

−

bε2 cos 4θ 4a3

(4)

In the figure below (to the left) I show a constant energy contour, which differs visible from a circle. It is obtained by a numerical evaluation of (3) (Mathematica program) in the case of a = b = 1 and ε = 2 (assuming dimensionless quantities). The thin line shows the average length of k(ε). The figure to the right show the corresponding |∇k εk | as a function of the angle θ. The gradient, and hence the velocity vk = ∇k εk /, is perpendicular to the constant energy contour. Notice that |v | is smallest along the 11 directions, where |k(ε)| has its maxima. k

15


2) The Boltzmann equation in the relaxation-time approximation, (17.17)-(17.18), dg g−f ∂g ˙ ∂g ˙ ∂g =− = + r · +k· dt ∂t ∂r τ ∂k

k˙ · ∂g = − g − f τ ∂k

⇒

(5)

when considering the steady state of a uniform system. The fields are assumed to = (E, 0, 0) and B = (0, 0, B), and the semiclassical equation of motion is be E

˙ + 1 v × B , k = −e E c k

1 ∂εk vk = r˙ = ∂k

(6)

Introducing this in the right-hand part of (5) we get

e 1 · − E + vk × B c where

∂(g − f ) ∂f + ∂k ∂k

=−

g−f τ

∂f ∂f ∂εk ∂f = = vk ∂ε ∂k ∂ε ∂k k k

(7)

(8)

and the linearized version of (7) is This gradient is perpendicular to v × B, k

e ∂f · ∂(g − f ) = − g − f − vk × B (9) k ∂ε c τ ∂ k e ∂(g − f ) · , is of Since (g − f ) is going to scale with E, the term neglected, − E ∂k = (E, 0, 0) and B = (0, 0, B) in (9), we finally get second order in E. Inserting E · v −eE k

∂f eB ∂ ∂ = v − vx −evx E ∂ε c y ∂kx ∂ky

k

1 − (g − f ) τ

(10)

3) With the assumption of g − f = ky F (ε), we find vy

∂(g − f ) ∂(g − f ) dF ∂ε dF ∂ε − vx = vy ky − vx F − vx ky ∂kx ∂ky dε ∂kx dε ∂ky dF dF v − vx F − vx ky v = −vx F = vy ky dε x dε y

(11)


16

When introducing this result in (10) and neglecting 1/τ , we get −evx E

∂f eB [−vx F (ε)] = ∂ε c

⇒

F (ε) =

k

cE ∂f B ∂ε

(12)

k

The current in the y direction is then jy = −e

ecE [dk]vy (g − f ) = − B

ecE =− B =

ecE B

dkx

∂f [dk]vy ky ∂ε

k

∂εk

ecE ∂f ky dky Dk =− ∂(ky ) ∂ε B

dkx dky Dk f =

ecE B

k

dkx dky Dk ky

necE 1 [dk]f = =− E, B RB

∂f ∂ky

R=−

(13) 1 nec

when performing the y integration by parts, where Dk = 2/(2π)2 is a constant and the boundaries of the integral is the boundaries of the first Brillouin zone. (13) is the usual high-field result for the off-diagonal conductivity σyx = −1/(RB).

Solutions to the problems in Chapter 22 and 23 22.2 Faraday rotation The combination of the semiclassical equation of motion and the Drude relaxation˙ time model leads to the following equation of motion for the electrons (when k = m∗v˙ = m∗¨r) ˙ e − r − e E ¨r = − ∗ r˙ × B m c m∗ τ

or

e r˙ =− e E ¨r + + ∗ r˙ × B τ m c m∗

(1)

The (applied) magnetic field is assumed along zˆ. The electrical field vector, due to an incident light wave, is assumed to be circular polarized in the plane perpendicular to z = (0, 0, B), =E ± = E0 (ˆ x ± iˆ y ) e−iωt (2) B E If the light wave propagates in the positive z direction, i.e. E ∝ ei(qz−ωt) , then the plus (minus) sign in (2) corresponds to the left (right) circular polarization. In the case of the right circular polarization, the end point of the E-vector is making a right-handed screw line in the direction of propagation at a certain time t (see for instance Griffiths, “Introduction to electrodynamics”, p. 374). (a) Introducing the expressions for the fields in (1), the equation may be solved, with respect to r˙ , by assuming x ± iˆ y ) e−iωt r˙ = A± (ˆ

(3)

Because (ˆ x ± iˆ y ) × zˆ = −ˆ y ± iˆ x = ±i(ˆ x ± iˆ y ), we get from (1)

eB 1 e (−iω) + ± i ∗ A± (ˆ x ± iˆ y )e−iωt = − ∗ E0 (ˆ x ± iˆ y )e−iωt τ m c m

(4)

or

eE 1 eB A = − ∗0 (−iω) + ± i ∗ m τ m c ±

−1

=−

eE0 τ , ∗ m 1 − iωτ ± iωc τ

ωc =

eB (5) m∗ c

17

Problems in Chapter 22 and 23 (“optical properties”)

where ωc is the cyclotron frequency (section 21.2). The current density and the conductivity tensor are determined from j = −ner˙ ,

j = σ E

(6)

and defining j ± = j (ˆ y ), then The calculations show that r˙ = (A± /E0 )E 0 x ± iˆ + + − + + − +σ E = σ E and j = σ E + + σ E − = σ− E − . The j = σ++ E +− −+ −− current has the same circular polarization as the electric field, i.e. the conductivity tensor is diagonal with respect to the choice of sign, σ+− = σ−+ = 0. The diagonal components are (σ++ ≡ σ + and σ−− ≡ σ − ) σ ± = −ne

A± σ0 , = E0 1 − iωτ ± iωc τ

σ0 =

ne2 τ m∗

(7)

(b) In the case of ωτ 1 and ω ωc , we get

σ0 i σ0 ω 1+ σ = = ∓ c 1 − iωτ ± iωc τ (−iωτ ) ωτ ω ±

−1

iσ i ω 0 1− ± c ωτ ωτ ω

(8)

(c) The dielectric constant, (20.14), is

ωp2 4πi ± i ω =1+ σ 1− 2 1− ± c ω ω ωτ ω ±

(9)

where ωp is the plasma frequency introduced by (20.32) or (23.7) ωp2 =

4πne2 4πσ0 = ∗ m τ

(10)

The index of refraction, (20.18), is

n ˜ ± = n± + iκ± =

±

ωp2 i ω 1− 2 1− ± c , 2ω ωτ ω

ω ωp

(11)

The Faraday rotation θ of the polarization vector per unit length along the sample is then found to be θ=

ωp2 ω ωp2 ωc ω ω ·2 2 c = n− − n+ = 2c 2c 2ω ω 2c ω 2

(12)

is along the Notice, that the sign of θ is determined by the sign of ωc , or B, i.e. if B negative z direction then θ is negative. If the active carriers are holes rather than electrons, we may perform exactly the same calculation except that e is replaced by −e in all expressions (m∗ is then the positive hole mass), hence the sign of θ, like the Hall voltage, depends on the kind of carriers. Derivation of the expression for the Faraday rotation angle: Assuming the incident light wave to be linearly polarized along the x axis at z = 0, then we want to find the angle of rotation of the polarization vector in the xy plane as a function of z. 1 E(z)/E x + iˆ y )ei(q+ z−ωt) + 12 (ˆ x − iˆ y )ei(q− z−ωt) 0 = 2 (ˆ

(13)


18

At z = 0, the polarization vector is parallel to x ˆ , E(0)/E ˆ e−iωt . Introducing 0 = x 1 1 ψ = 2 (q+ + q− )z and ∆ψ = 2 (q+ − q− )z, or q+ z = ψ + ∆ψ and q− z = ψ − ∆ψ, then we get E(z)/E 0 =

1 x 2 (ˆ

+ iˆ y )ei∆ψ + 12 (ˆ x − iˆ y )e−i∆ψ ei(ψ−ωt)

(14)

The polarization vector at z is determined by the factor in the square bracket, which is found to be 1 x + iˆ y )(cos ∆ψ 2 (ˆ

+ i sin ∆ψ) + 12 (ˆ x − iˆ y )(cos ∆ψ − i sin ∆ψ) = x ˆ cos ∆ψ − yˆ sin ∆ψ

This expression shows that the polarization vector at z is making the angle θz = −∆ψ = − 12 (q+ − q− )z with the x axis. According to (20.18) q± = ωn± /c, and the rotation angle per unit length, along the direction of light propagation, is θ = − 12 (q+ − q− ) =

ω n− − n+ 2c

(15)

In the case where the carriers are electrons, then θ > 0, which sign corresponds to a counter-clockwise or left-handed rotation of the polarization vector along z. 23.3 Helicon waves (a) From the Maxwell equations (23.71) and (23.72) we find 2 2 = ∇ × ∂B = 1 ∂ E + 4π ∂ j = 1 ∂ E + 4πσ ∂ E −c ∇ × ∇ × E ∂t c ∂t2 c ∂t c ∂t2 c ∂t

(1)

Assuming when utilizing the linear relation j(ω) = σ(ω)E(ω). =E 0 ei(k·r−ωt) , E

k = (0, 0, k),

k0 ≡

ω c

(2)

we get from the left- and right-hand sides of (1)

2 + 4πσ (−iω)E = 1 (−iω)2 E = −ω −c (ik) × (ik) × E c c c

2

= −ω E E c (3) when introducing the dielectric tensor , (20.14). Leaving out the common phase factor, we finally get k × k × E + k2 E = 0 (4) 0 0 0 4πσ 1+i ω

The conducting sample is assumed (b) Symmetry analysis of the relation j = σ E: to be isotropic (in the shape of a sphere, if surface effects are of importance). In the = Bˆz , the sample, or alternatively the coordinate presence of a magnetic field B axes, may be rotated any angle around the z axis without changing the coordinate If representation (the matrix equation) for the tensor relation between j and E. the coordinate axes are rotated by an angle of 90 degrees around the z axis, then (jx , jy , jz ) is renamed (−jy , jx , jz ) and (Ex , Ey , Ez ) is replaced by (−Ey , Ex , Ez ), whereas the components σαβ determined by the sample are kept unchanged: ⎧ ⎨ jx = σxx Ex + σxy Ey + σxz Ez

j =σ E +σ E +σ E

⎩ jy = σyx Ex + σyy Ey + σyz E z z zx x zy y zz z

→

⎧ ⎨ −jy = −σxx Ey + σxy Ex + σxz Ez ⎩

jx = −σyx Ey + σyy Ex + σyz Ez jz = −σzx Ey + σzy Ex + σzz Ez

(5)

19

Problems in Chapter 22 and 23 (“optical properties”)

A comparison of the two sets of relations then shows that σxx = σyy , σxy = −σyx , and σzx = σzy = σxz = σyz = 0. [Extension of the symmetry arguments: At zero field, the isotropic sample is insensitive to a rotation around any arbitrary direction and σαβ = σxx δαβ , which result shows that σxy → 0 when B → 0. A reflection of the system with respect to, for along zˆ is reversed, whereas instance, the yz plane implies that the axial vector B it is the x components of the polar vectors j and E which change sign. Because of this different behaviour of the polar and axial vectors it is straightforwardly shown is reversed, whereas nothing happens with the diagonal that σxy changes sign if B elements. This symmetry element is in accordance with the general Onsager relation σαβ (−B) = σβα (B). Hence, in the case of an isotropic sample the diagonal components may only involve terms quadratic (even powers) in B, whereas the off-diagonal ones are linear (of odd powers) in B.] These symmetry relations are consistent with the results obtained in the previous problems 17.9 and 22.2, and they apply as well for the dielectric tensor , or ⎛

xx = ⎝ −xy 0

xy xx 0

⎞

0 0 ⎠ zz

(6)

+ k(k · E ) = −k2 E ), the Introducing this in (4) in combination with k × (k × E 0 0 0 result is ⎛ ⎞⎛ 0 ⎞ −k2 + xx k02 xy k02 0 Ex 2 + k2 ⎝ − k 2 ⎠ ⎝ Ey0 ⎠ = 0 −k 0 (7) xy 0 xx 0 Ez0 0 0 zz k02 , if the characteristic determinant vanwhich only has a non-zero solution for E 0 ishes, or if 2 (8) zz (k/k0 )2 − xx + 2xy = 0 We are going to utilize the solution of problem 22.2 in the following. Here σ ± is ± in the xy plane, or defined by the vector relation j ± = σ ± E jx + ijy =

+ σ (Ex + iEy )

(9)

σxx Ex + σxy Ey + i(−σxy Ex + σxx Ey ) = (σxx − iσxy )(Ex + iEy )

showing that σ + = σxx − iσxy and correspondingly, (9) in the solution of 22.2, ± = xx ∓ ixy

⇒

xx = 12 (+ + − ),

xy = 12 (+ − − )i

(10)

The latter relations reduce (8) to the diagonal form

zz (k/k0 )2 − +

(k/k0 )2 − − = 0

(11)

(c) The situation is exactly the one considered in problem 22.2, and the result found there is [combining (8) and (9) in the solution of 22.2] ωp2 4πi σ0 τ =1+i ω 1 − iωτ ± iωc τ ω 1 − iωτ ± iωc τ ωp2 1 ωp2 i 1 = 1 − , ω ˜ =ω+ =1+ ω iτ1 − ω ± ωc ω ω ˜ ∓ ωc τ

± = 1 +

(12)


20

or finally ± = 1 −

ωp2 1 ± (ωc /˜ ω) , ωω ˜ 1 − (ωc /˜ ω )2

ωc =

eB , m∗ c

4πσ0 4πne2 = τ m∗

ωp2 =

(13)

which result agrees with the expression for xx = (+ + − )/2 given by (23.82). (d) The atomic density (front page) of aluminium metal is 6.02 · 1022 cm−3 . According to Table 6.1, the number of conduction electrons per atom is 3, i.e. the electron density in Al3+ -metal is n = 18.06 · 1022 cm−3 . The cyclic plasma frequency is ωp = 5.64(15) · 1015 rad/s, when n = 1022 cm−3 and m∗ = m, see (23.3). [Incidentally, the unit of a cyclic frequency is not Hz but rad/s or just s−1 . The unit Hz (hertz) is reserved for the case of cycles per second, ν = ω/(2π).] In the case of Al-metal, ωp = 24 · 1015 s−1 ,

νp = 3.8 · 1015 Hz,

ωp = 16 eV

(14)

The cyclotron frequency may be expressed in terms of the fundamental magnetic quantity, the Bohr magneton: µB =

e = 5.788383 · 10−9 eV/G 2mc

⇒

ωc =

m eB = 2 ∗ µB B m∗ c m

(15)

Assuming m∗ = m and B = 1 T = 10 kG = 104 G, we get νc = 2.8 · 1010 Hz,

ωc = 0.12 meV,

ωc = 18 · 1010 s−1

(16)

The numbers show that ωc 10−5 ωp ωp . In the case of ω ≈ ωc , the fraction in the expression for ± is of the order of 1010 , and the adding of one to this fraction has no effect at all. (e) Assuming ω ωp we get from equation (12) ± = −

ωp2 ω(ω ∓ ωc + τi )

−

ωp2 , ω(ω ∓ ωc )

ωc τ 1

(17)

In most cases ± < 0, which implies a purely imaginary refraction index n ˜ = √ and therefore no light propagation. However, there is one situation, where electromagnetic waves with ω ωp are capable to propagate through the sample, which one is determined by the solution + , when ω < ωc (or − if the carriers are holes). Hence, when placing a metal like aluminium in a strong magnetic field, it is possible for low-frequency light to propagate through the metal, but low temperature and high purity is demanded in order to secure that the damping is small, ωc τ 1. The dispersion relation for these “helicon” waves is determined by (11) and (17): 2

ωp2 ω(ωc − ω)

ω ωc (ck)2 2c (ck)2 , 2 2 ωp + (ck) ωp

ωp c (18) showing that the dispersion is not linear but quadratic in k in the long wavelength limit. The name of the waves is a derivative of “helix” and refers to the circular polarization of the waves (+ corresponds to left circular polarization) – the polarization vector of the helicon wave is forming a helix along the z direction. 2

k =

k02 +

=

ω c

⇒

ω=

k

21

Problems in Chapter 24 (“ordering”)

The helicon resonance is related to the Hall effect, and the result may be written in a simple form in terms of the Hall coefficient R = −(nec)−1 : ω=

B|R| (ck)2 4π

(cgs-Gaussian units)

(19)

In the experiments of, for instance, R. Bowers, C. Legendy and F. Rose, Phys. Rev. Lett. 7, 339 (1961), wavelengths of the order of 1 cm are considered. In the case of Al-metal, λ = 1 cm corresponds to a helicon frequency of ν = 1.77 Hz at B = 10 kG. In sodium (n = 2.54 · 1022 cm−3 ) this frequency is 12.3 Hz. Bowers et al. observed a standing wave resonance in a Na sample of length l = 4 mm at 32 Hz and 10 kG. The resonance condition is l = λ/2, and our simple model predicts λ/2 to be about 3.1 mm at the same frequency and field.

Solutions to the problems in Chapter 24 24.4 Mean field theory The antiferromagnetic two-dimensional Ising model on a square lattice. In this lattice, the number of nearest neighbours is z = 4. The Hamiltonian is supposed to be

H=−

R R

J σR σR −

R

HµB σR =−

1 J σR HµB σR σR − , 2 RR

J < 0 (1)

R

where H = E in the usual representation of σR = ±1. Because J is negative, the and R ground state is not the ferro- but the antiferromagnet. In the sums, R are nearest neighbours, and the antiferromagnetic ground state is assumed to be determined by the mean field values (σR , σR ) =

⊂ sublattice A R ⊂ sublattice B R

(σ↑ , σ↓ ), (σ↓ , σ↑ ),

(2)

The two sublattices of the square lattice: A shown by filled and B by open circles. The fig belongs ure shows the case, where the site at R to the A-sublattice, in which case σR = σ↑ . , then Its four nearest neighbours at positions R all belong to the B sublattice and σR = σ↓ .

(a) In the mean-field (MF) approximation we get, when inserting (2) in (1), MF

E= − =−

1 J σR σR σR + σR σR − σR σR − HµB 2

RR

R R

J σR σR − HµB

N/2

=−

R⊂A

R

R

σR +

zJ σ↓ + µB H σR −

1 2 N zJ σ↑ σ↓

N/2

R⊂B

(3)

1 zJ σ↑ + µB H σR + 2 N zJ σ↑ σ↓


22 !

Because of the MF-approximation, the total energy E = R ER is the sum of the individual contributions of each site, and the partition function of the total system Z=

all

−β E

e

=

E

"

R

all

−β ER

e

=

"

ER

R

ZR

(4)

is reduced so to become the product of the partition functions ZR for the individual belongs to the sublattice A sites, see for instance (6.40). For the sites, where R (↑), we get, when omitting the constant energy term 12 zJ σ↑ σ↓ ,

Z↑ =

e−β [−(zJσ↓ +µB H)σ] =

σ=±1

eβ (zJσ↓ +µB H )σ = 2 cosh zJ σ↓ + µB H

(5)

σ=±1

which implies that σR = σ↑ =

1 σ eβ (zJσ↓ +µB H )σ = tanh β(zJ σ↓ + µB H) Z↑ σ=±1

(6)

Notice that the omission of a constant energy term does not affect the calculation of a thermal expectation value. Z↓ and σ↓ are determined equivalently

Z↓ = 2 cosh zJ σ↑ + µB H ,

σ↓ = tanh β(zJ σ↑ + µB H)

(7)

and the (total) free energy is

F=−

N kB T ln Z↑ + ln Z↓ + N 12 zJ σ↑ σ↓ = 2

# $ N N kB T ln 2 cosh (zJ σ↓ + µB H) + ln 2 cosh (zJ σ↑ + µB H) + zJ σ↑ σ↓ 2 2 (8) The constant energy term neglecting in the partition functions adds to F.

−

(b) The self-consistent equations determining σ↑ and σ↓ are derived above. (c) In the paramagnetic phase, T > TN , σ↑ and σ↓ vanish in the limit H → 0. Expanding tanh to leading order, tanh x x, we get from (6) and (7)

⎫

σ↑ = tanh β(zJ σ↓ + µB H) β(zJ σ↓ + µB H) ⎪ ⎬ ⎪

σ↓ = tanh β(zJ σ↑ + µB H) β(zJ σ↑ + µB H) ⎭

σ↑ = σ↓ =

µB Hβ 1 − zJ β

(9)

The expectation values σ↑ and σ↓ are equal and are proportional to the field. The magnetization is M = (N/V )µB (σ↑ + σ↓ )/2 and the susceptibility is defined as χ = M/H in the limit of zero field, and we find χ=

µ2B N µ2B N ≡ , V kB T − zJ V kB (T − Θ)

Θ=−

z|J | = −TN kB

(10)

The paramagnetic Curie, or Curie–Weiss, temperature Θ is defined by (24.41), χ ∝ (T − Θ)−1 . In the present system Θ is negative (J is negative), and |Θ| is, as we shall see, the same as the “Néel temperature”, TN , below which the system is antiferromagnetically ordered at zero field.

23


(d) The Néel temperature is the temperature below which the self-consistent equations for σ↑ and σ↓ have a non-zero solution at zero field. The solution is obtained by assuming σ↑ = −σ↓ , and H = 0, in which case we get

σ↑ = tanh β(−z|J |)(−σ↑ ) = tanh(βz|J |σ↑ ) = tanh(ασ↑ )

(11)

This equation is the same as considered in the case of the ferromagnet, (24.56), and it has a non-zero solution, when the coefficient α ≥ 1. The Néel temperature TN is the temperature at which α = z|J |β = 1, or TN = z|J |/kB . The paramagnetic susceptibility, which diverges at the transition, is obtained by applying a field with the same symmetry as the ordered phase. The application of a “staggered” field, where HR = +H at the A sites and −H at the B sites determines straightforwardly (using σ↑ = −σ↓ ) the staggered susceptibility to be

χstag =

N N µB (σ↑ − σ↓ ) µ2B = V 2H V kB (T − TN )

(12)

which diverges at TN . (e) Spin-flip transition: In the ordered phase, the application of a small field along z, in the “up” direction, does not change σ↑ = 1 or σ↓ = −1. However, if the field becomes sufficiently large, the spins antiparallel to the field are going to flip so to become parallel to the field. This first-order spin-flip transition happens, when the Zeeman-energy gain is able to compensate for the loss of exchange energy. Introducing dimensionless quantities, then

−σ↓ + h , σ↑ = tanh t

−σ↑ + h σ↓ = tanh , t

h=

µB H , z|J |

t=

T TN

(13)

In the limit of t → 0, then the introduction of σ↑ = 1 in the second equation implies σ↓ = −1 if 1 > h > 0 and σ↓ = 1 if h > 1. The first equation then predicts σ↑ = 1 in both cases, hence assuring that this is the self-consistent solution. Using ln[2 cosh x] = ln[ex + e−x ] → ln e|x| = |x| for x → ±∞, the free energy is found to be determined by

t F −σ↓ + h =− f= ln 2 cosh N z|J | 2 t t →− 2

)

)(

−σ↑ + h + ln 2 cosh t

−σ↓ + h )) −σ↑ + h )) 1 +) ) − 2 σ↑ σ↓ = t t

(

− 12 σ↑ σ↓

− 12 , 01

(14) Hence, the free energies of the antiferromagnetic phase f = −1/2 and of the spinflipped phase f = −h + 1/2 are equal at the transition at h = 1. At a finite temperature the spin-flip transition is smeared out due to thermal effects. The figure below shows the numerical calculated “magnetization”, (σ↑ + σ↓ )/2, as a function of the field h at various values of temperature t.


24

1.2 t 0 0.2 0.4 0.6 0.8 1.0

Averaged spin

1.0 0.8 0.6 0.4 0.2 0.0 0.0

0.4

0.8 1.2 Magnetic field (h)

1.6

24.6 Superlattices Model of Cu3 Au – In the fcc lattice, the sites in the centers of the six faces are named As and the 8 corners are named Bs as indicated on the figure by, respectively, black and white spheres. There is 8/8 = 1 Bs site and 6/2 = 3 As sites per unit cell. √ The distance between nearest neighbours is a/ 2, and the coordination number is z = 12. The kind of nearest neighbours of the two kind of sites are

As -site: 8 As sites and 4 Bs sites. Bs -site: 12 As sites. The correct solution to the equations (24.62) and (24.63) is not (24.64), but f (σR , σR ) =

BB − AA 2AB − AA − BB (σR σR + σR ) − σR 4 4

(24.64 )

when leaving out the constant term C1 = (2AB + AA + BB )/4. This modification affects the definition of the effective parameters in (24.67), which should read µB H = µ −

BB − AA z, 2

J=

2AB − AA − BB 4

(24.67 )

(a) In the present system we have a 3:1 mixture of A and B atoms. The effective Hamiltonian equals the energy minus the chemical potential times the difference between the numbers of B and A atoms, is (leaving out the constant contribution)

H=E−µ

R

σR = −J

R R

σR σR − µB H

R

σR

(1)

We are going to use the same choice as Marder, that σR = 1 or −1 signifies a site occupied by, respectively, a B or an A atom. With this choice, the effective energy parameters are those given by (24.67 ) above. In the mean-field approximation MF

H = −J

R R

σR σR + σR σR − σR σR − µB H

R

σR =

R

HR

(2)

25


where R R

1 σR σR σR σR +σR σR = σR + σR σR = 2 RR

=

N

s

3N s A

(nn) R

R

σB 12 σA + s

s B

σR

σA 8 σA + 4 σB s

s

s

(3) (σB ), if R = As

Here N is the number of unit cells (or Bs sites), and σR = σA s s s ) is the position of an As (Bs ) site. The (grand) partition function of the total (B system is the product of the individual partition functions for each site, because ! H= R HR , and in the case of an As site (omitting the constant energy term)

ZAs =

β [J (8 σ +4 σ )+µB H ]σ As

e

Bs

(4)

σ=±1

implying that σA = s

β [J (8 σ +4 σ )+µB H ]σ 1 As Bs σe ZAs σ=±1

#

= tanh β 8 J σA + 4 J σB + µB H s

and equivalently

(5)

$

s

#

σB = tanh β 12 J σA + µB H s

$

(6)

s

(b) σR = +1 for the B atoms and the number of these atoms is equal the number of unit cells N , whereas the number of A atoms is 3N . These conditions imply ⎧ N − 3N = −2N ⎪ ⎪ ⎪ ⎨ N 3N σR = ⎪ σB σA ⎪ s + s = N σB s + 3N σA s ⎪ R ⎩ s B

⇒

σB = −2 − 3 σA s

s

s A

(7) (c) The chemical potential, or effectively µB H, has to be adjusted so that the two equations (5) and (6) are in accordance with the relation σB s = −2 − 3 σA s . The effective Zeeman term may be eliminated:

−1 tanh−1 σA − tanh σB = β 8 J σA + 4 J σB + µB H

s

s

s

s

−β 12 J σA + µB H = 4βJ σB − σA = −8βJ 1 + 2σA s

s

s

(8)

s

Introducing an effective order parameter Q and J = −|J |, we finally get −1

f (Q) ≡ tanh

Q+1 2

−1

+ tanh

3Q − 1 2

= 8β|J |Q,

Q = −1 − 2σA (9) s

In the disordered phase both the As and the Bs sites are occupied by an A atom 3 1 1 with the probability 34 or σA s = σB s = (−1) 4 + (+1) 4 = − 2 . This result is in accordance with (7) and Q = −1 − 2σA = 0. In the completely ordered phase at s


26

zero temperature, all A (B) atoms are placed on the As (Bs ) sites implying that σA = −1 and σB = 1 and hence Q = 1. s

s

The numerical solution of (9) is discussed in a Mathematica program. The system orders at a first-order transition, when f (Q) = Q f (Q) [the line y = αQ with α = f (Qc ) is parallel to the tangent of f (Q) at Q = Qc , and this line just touches f (Q) at Q = Qc , when f (Qc ) = αQc . This then becomes a solution to (9) if choosing α = 8β|J|]. The solution is Q = Qc = 0.3455 and defining the effective temperature scale t = (8β|J|)−1 then Q jumps from zero to Qc at the temperature t = tc = 1/f (Qc ) = 0.4143. The calculated variation of Q as a function of t is shown in the figure.

Solutions to the problems in Chapter 25 and 26 25.2 Classical electrons in a magnetic field The spin degree-of-freedom does not occur in classical physics. The elementary particles do neither exist, but accepting their existence, they only interact with the electromagnetic fields because of their charges, and with the gravitational field because of their masses. The classical Hamiltonian H of a many-body system (see for instance Goldstein) is a function of the independent canonical variables (ri , pi ) of all the particles. The canonical impulse pi , independent of ri , is defined in terms of the Lagrangian L = L(ri , r˙ i , · · ·) as pi = ∂ L/∂r˙ i (or, in terms of the Hamiltonian ri ). theory, as the generator of an infinitesimal translation ri → ri + δ (a) If the external constraints (fields) are independent of time, the Hamiltonian is equal the total (kinetic plus potential) energy of the system. In this case the classical partition function is Z = e−β F =

1 hn

dr1 dp1 · · · drn dpn e−β H(r1 ,p1 ,···,rn ,pn )

(1)

only contributes (b) In the presence of a time-independent electromagnetic field, E to the potential energy as determined by the scalar potential. The magnetic field only affects the velocities of the particles via the vector potential A = A( r ), B where B = ∇ × A. For the ith particle with mass mi and charge qi , the canonical impulse is q pi = mir˙ i + i A c

⇒

2

1 q Ti = 12 mi (r˙ i )2 = pi − i A 2mi c

(2)

[The magnetic field needs not to be uniform in space, but only to be constant in time – the case of a spatial uniform field is discussed in the next problem 25.4].

27

Problems in Chapter 25 and 26 (“magnetism”)

(c) The canonical impulse pi appears in the Hamiltonian only via the kinetic energy Ti . Assuming H in (1) to be the Hamiltonian function of the system, when the magnetic field is zero (but including the potential energy contributions of the electric field), then the introduction of the magnetic field implies that

H(r1 ,p1 , · · · ,rn ,pn ) → H r1 ,p1 −

q1 q A(r1 ), · · · ,rn ,pn − n A( rn ) c c

(3)

The partition function Z in the presence of the magnetic field is then

q 1 r ),···,r ,p − qn A( r ) −β H r1 ,p1 − c1 A( n n n 1 c d r d p · · · d r d p e 1 1 n n hn 1 = n dr1 dp1 · · · drn dpn e−β H(r1 ,p1 ,···,rn ,pn ) = Z(0) h

= Z(B)

(4)

The integration with respect to the canonical impulse pi may be performed before q ri ), the ri -integrations. Introducing the following change of variables pi = pi − ci A( then dpi = dpi , since A(ri ) is independent of pi . This transformation of variables in the partition therefore leads to the same integral as in (1). The elimination of B function means, for instance, that the magnetic susceptibility of a classical equilibrium system is zero. This result is called the Bohr–van Leeuwen theorem. Notice, that it is important for the argumentation that the system is in equilibrium (the basis for the use of the partition function). 25.4 Quantum electrons in a magnetic field is The Hamiltonian of an electron with charge −e in a magnetic field B

1 H= p + 2m

e A c

2

(1)

The choice of the “Landau gauge” = (0, B x, 0) A

⇒

=∇×A = (0, 0, B) , B

=0 ∇·A

(2)

in the z direction. i.e. this vector potential corresponds to a spatial uniform field B is zero, the two quantum mechanical operators A and p Since the divergence of A commute, p = (/i)∇ in the r -representation. Introducing this choice of gauge in (1), we get

2 1 eB 2 2 x + pz H= p + py + (3) 2m x c The wave function of the electron is assumed to be ψ(r ) = u(x) ei(yky +zkz )

(4)

and the eigenvalue equation is

i(yky +zkz )

Hψ = e

2

eB 1 x p2x + ky + 2m c

2

+ (kz )

u(x)

(5)

When the common phase factor is deleted, this eigenvalue equation is reduced to one, which only involves a determination of u(x), and the equation may be written:

p2x H u(x) = + 1 m ωc2 (x − x0 )2 + εz u(x) 2m 2

(6)

Condensed Matter Physics 2 where ωc =

eB , mc

x0 = −

ky , mωc

28

εz =

2 kz2 2m

(7)

Equation (6) is the eigenvalue equation for the one-dimensional harmonic oscillator, displaced to be centered around x = x0 and shifted in energy by the constant (with respect to x) energy εz . Hence, the final eigenstates are classified by (ky , kz ) and the occupation number ν, and the energy eigenvalues are

1 2

εν,ky ,kz = ν +

ωc +

2 kz2 , 2m

ν = 0, 1, 2, · · ·

(8)

This is the result (25.48) utilized in the analysis of the Landau diamagnetism.

26.2 Ferromagnetic ground state We are going to discuss the ferromagnetic ground state of the Heisenberg model for an arbitrary integer/half-integer value of S. The Heisenberg Hamiltonian is

H=−

ll

· S = − 1 Jll S l l 2

l=l

·S Jll S l l

(1)

according to (26.21) in Marder. The possibility of Jll = 0 may be excluded, since such a term would contribute only by a constant term Jll S(S + 1) to the energy. fulfill the usual spin-commutator relations The components of S l [Slx , Sl y ] = i δll Slz ,

[Sly , Sl z ] = i δll Slx ,

(a) Defining the following operator:

Sz =

[Slz , Sl x ] = i δll Sly

(2) (3)

Slz

l

in (1) is ·S then the commutator of this operator with a single term S l l · S ] = [ S , S ]·S + S · [ S , S ] [ Sz , S l l z l l l z l (4)

= [Slz , Slx ] Sl x + [Slz , Sly ] Sl y + Slx [Sl z , Sl x ] + Sly [Sl z , Sl y ] = i Sly Sl x − i Slx Sl y + i Slx Sl y − i Sly Sl x = 0

showing that [ Sz , H ] = 0. The Sz -representation for each site is defined by * ! * Slz |ml = ml |ml , where ml = −S, −S+1, . . . , S, and Sz l |ml = ( l ml ) l |ml . Most of these eigenstates are highly degenerated except for the two “singlets”, * * * l |ml = ±S = l | + S or l | − S . Because Sz and H commute the two non-degenerate eigenstates of Sz are also eigenstates to H, and

H

"

|ml = ±S = − 12 N J (0)S 2

l

"

J (k) =

|ml = ±S ,

l

l

Jll e−ik·(rl −rl ) (5)

and using S2 = (S +S )2 = S 2 + S 2 + 2S ·S +S (b) Defining S = S R R R R R

Ψ| S2 |Ψ =

R

R

· S |Ψ S(S + 1) + S(S + 1) + 2Ψ|S R R S(S + 1), S = 0, 1, · · · , 2S

R

(6)

|Ψ attains its maximum value, when S = 2S, and ·S which shows that Ψ|S R R

|Ψ} = 1 [2S(2S + 1) − 2S(S + 1)] = S 2 ·S Max{Ψ|S 2 R R

(7)

29


(c) The eigenvalue of H in (5) for the fully polarized states requires that each pair of operators in (1) contributes by their maximum. Hence, if all exchange constants Jll ≥ 0, then the eigenvalue in (5) is the lowest possible value, i.e. the two fully polarized states are the two degenerate ground states. The lowest exited states are linear combination of the states, where all ml = S except that ml = S − 1 for one particular site. Naming the product of these states for |l, where l is the site at which ml = S − 1, then it is straightforward to show that the eigenstates of H constructed from these states are the Fourier transforms of |l. These excited states are characterized by the wave vector k of the Fourier transformation, and the energy difference between one excited k-state and the ground state is the spin wave energy, S[J (0) − J (k)]. Hence, these lowest excited states, the one-magnon states, are exact solutions to the Heisenberg Hamiltonian – but this is no longer the case if proceeding to the two-magnon states, where the eigenvalue of Sz is N S − 2.

26.4 Diagonalizing spin waves The result derived in (26.62) may be written in a reduced form: ˆ= h =

1 + 1 H + N z|J |S 2 = 2a ak + a+ a+ + ak a−k cos(k · δ) k k −k 2z|J |S 2z

k

a+ ak k

+

1 2

δk

a+ a+ k −k

1 Bk = cos(k · δ) z

+ ak a−k Bk ,

(1)

δ

The operators are defined by (26.51) in terms of al and its hermitean conjugate a+ l , which are the annihilation and creation operators, also named Bose operators, of a harmonic oscillator at the site l. These operators obey the Bose commutator + + relations [al , a+ l ] = δll and [al , al ] = [al , al ] = 0, which imply 1 ik·rl + −ik ·rl 1 1 i(k−k )·rl al e , al e = δll eik·rl −ik ·rl = e k N ll N ll N l (2) or that the Fourier transforms of the Bose operators, diagonal with respect to the site index, are Bose operators in reciprocal space [ak , a+ ] =

[ak , a+ ] = δkk , k

[ak , ak ] = 0 ,

[a+ , a+ ] = 0 k

k

(3)

which are diagonal with respect to the wave vector index (notice that the wave vector k is assumed to be a discrete variable). The Hamiltonian (1) is not diagonal in the number representations of the harmonic oscillators, since it involves a+ a+ + ak a−k . One way to solve an eigenvalue problem is to make a canonical k −k transformation of the Hamiltonian to one for which the eigenvalue equations have been solved. In the present case, we want to find a new Bose operator γ , in k ˆ is ˆ only depends on the number operators γ + γ , in which case h terms of which h k

k

“diagonal”. This procedure is called the Bogoliubov transformation: ak = uk γk + vk γ + , −k

a+ = u∗k γ+ + vk∗ γ−k = uk γ+ + vk γ−k k

k

k

(4)


30

The last equation indicates that uk and vk are assumed to be real functions. This is a valid simplification in the present case, but it is not a general requirement. The demand that the new operators should also be Bose operators implies [ak , a+ ] = [uk γk + vk γ + , uk γ+ + vk γ−k ] = uk uk δkk − vk vk δ−k−k = δkk (5) −k

k

k

and, similarly, [a , a ] = u v δ − v u δ = 0. Hence, the fulfillment of the k k k k k−k k k −k k Bose commutator relations for γk requires u2k − vk2 = 1 ,

uk v−k − vk u−k = 0

(6)

Inserting (4) in (1) we get ˆ= h

uk γ+ + vk γ−k

×

u−k γ + −k

#

−k

k

k

uk γk + vk γ + + 12 Bk

+ v−k γk + uk γk +

uk γ+ + vk γ−k

k

vk γ + −k

u−k γ−k +

v−k γ+ k

$

(7)

Collecting corresponding terms and replacing −k with k in some of the terms (utilizing Bk = B−k and γk γ−k = γ−k γk ), then (7) is reduced to ˆ= h

2 + u2k + Bk uk v−k γ+ γk + v− k + Bk uk v−k γk γ k

k

#

k

$

1 2

+ 2uk vk + Bk uk u−k + vk v−k

γ+ γ + k −k

(8)

+ γk γ−k

In order to diagonalize this Hamiltonian we have to search for values of uk and vk , where the last term in (8) vanishes and the requirements of (6) are fulfilled. In order to simplify the search, we make the ansatz u = u , which is consistent k −k with the different conditions only if v−k = vk . In this case the equations are

2uk vk = −Bk u2k + vk2 ,

u2k − vk2 = 1

(9)

Introducing v2 = u2 − 1 in the squared version of the first equation, we get an k k equation of second degree in u2k . Determining the solution of this equation subject to the condition v2 > 0 or u2 > 1, we get k

u2k =

1 + ε

k

2ε

k

vk2 =

,

k

1 − ε

k

2ε

2uk vk = −

,

k

B

k

ε

,

εk =

k

1 − B2

(10)

k

According to the definition in (1), |B | ≤ 1 implying 0 ≤ ε ≤ 1. This is the same k k solution as given by equations (26.63) and (26.64) in Marder, since uk = cosh (αk ) and vk = sinh (αk ) imply u2k − vk2 = 1 and tanh (2αk ) = 2uk vk /(u2k + vk2 ) = −Bk . Introducing the solution (10) in (8) we finally get ˆ= h

1 2

k

εk + 1 γ+ γk + k

H = −N z|J |S 2 − z|J |S

εk − 1 γk γ+ =

k

k

Ek = 2z|J |S εk ,

1 2

1 − εk +

k

1 2

k

Ek γ+ γk k

γ+ γk = nk = [eβ Ek − 1]−1 k

εk − 1 + εk γ+ γk

k

(11)

31


The antiferromagnetic Néel state, where the spins of the two sublattices are either +S or −S is not an eigenstate of the Hamiltonian. The second term of H, discussed by Marder, shows that the ground state energy is smaller than that derived from the Néel state. Additionally, the magnitude |Sz (l)| is smaller than S at zero temperature. This “zero-point” reduction of the antiferromagnetic moment is determined by the site average of |0|Sz (l)|0| = S − 0|a+ l al |0, where |0 is the product ground state of the N independent harmonic oscillators, γk |0 = 0, ∆S =

1 1 0|a+ 0|a+ ak |0 = l al |0 = k N l N k

) , 1 2 1 1 − εk 1 + )) ) 0) uk γ+ + vk γ−k uk γk + vk γ + )0 = vk = k − k N N N 2εk k

k

(13)

k

which number is calculated [P.W. Anderson, Phys. Rev. 86, 694 (1952)] to be 0.078 in the case of a simple cubic lattice (z = 6).

Solution to HS’s problem 4 Localized atoms: heat capacity and magnetic susceptibility 1) In the case of non-interacting atoms, the total energy is the sum of the con! tributions from each site E = l El . This implies that the partition function is the product of the partition functions determined for each site (see also the note “Magnetic energy and domains”), and when the atoms are identical:

Z=

N "

Zl = Z N ,

Zl = Z =

e−βi = eβ∆ + 1 + e−β∆

(1)

i

l=1

The free energy is

F = −kB T ln Z = −N kB T ln Z = −N kB T ln eβ∆ + 1 + e−β∆

(2)

The internal energy is U = F + T S, where the entropy is ∂F 1 ∂Z = N kB ln Z + N kB T ⇒ ∂T Z ∂T 1 ∂Z 1 −βi 1 ∂Z = −N =N e U = F + T S = N kB T 2 Z ∂T Z ∂β Z i i

S=−

(3)

Introducing the population factor for the ith level (the probability that this level is occupied): pi ≡ !

e−βi Z

⇒

U =N

i

i pi = −N ∆

eβ∆ − e−β∆ eβ∆ + 1 + e−β∆

(4)

Notice, that i pi = 1. In terms of the population factors, the thermal expectation ˆ ˆ is A ˆ = ! i|A|ip value of any single site operator A i i , and the entropy is S = ! −N kB i pi ln pi .


32

2) In the limit kB T ∆ or β∆ 1, we have from (1) Z 1 + β∆ + 12 (β∆)2 + 1 + 1 − β∆ + 12 (β∆)2 = 3 + (β∆)2

⇒

F −N kB T ln 3 + (β∆)2 = −N kB T ln 3 1 + 13 (β∆)2 = −N kB T ln 3 −

S=−

∂F N ∆2 , = N kB ln 3 − ∂T 3kB T 2

U = F + TS = −

2N ∆2 3kB T

N ∆2 3kB T

(5) The heat capacity may be determined either as

2N ∆2 ∂U = C= ∂T 3kB T 2

∂S N ∆2 = T (−2) − C=T ∂T 3kB T 3

or

=

2N ∆2 3kB T 2

(6)

3) The derivative of F with respect to ∆ is 1 ∂Z 2N ∆ ∂F = −N kB T = −N kB T (βp1 + 0 · p2 − βp3 ) = N (p3 − p1 ) − (7) ∂∆ Z ∂∆ 3kB T The last equality is most simply obtained from F derived in (5). In the presence of a magnetic field H, applied along the z axis, the Hamiltonian is specified in terms of the Zeeman interaction

H=

Hl = −gµB H

l

Slz ,

S=1

(8)

l

The Hamiltonian of one of the sites is diagonal in the Sz -representation and the eigenenergies are ⎧ ⎨ |Sz = −1

|S = 0 ⎩ z |Sz = 1

⇒ i = gµB H = ∆ ⇒ i = 0 ⇒ i = −gµB H = −∆

(i = 3) (i = 2) (i = 1)

(9)

Hence the Hamiltonian (8) leads to the case considered when gµB H = ∆, and the derivative of F with respect to ∆ is proportional to the magnetization, or )

(gµB ) ∂F N 2 (gµB )2 1 ∂F =− H, M =− V ∂H V ∂∆ V 3kB T

M )) N 2 (gµB )2 χ= = H )H→0 V 3kB T (10) The susceptibility is calculated in the limit of H or ∆ → 0, hence the expression derived is valid at all T . Notice, that M may also be obtained from M V = gµB

Slz = N gµB

i|Sz |i pi = N gµB (p1 − p3 )

(11)

i

l

The result for χ in (10) is Curie’s law in the case of S = 1, see (25.31)-(25.32). 4) Defining x = β∆ the heat capacity derived from U in (4) is ∂U ∆ ∂U ∆2 ∂ C= =− = N ∂T kB T 2 ∂x kB T 2 ∂x =N

∆2 eβ∆ + e−β∆ + 4 kB T 2 (eβ∆ + 1 + e−β∆ )2

→

ex − e−x ex + 1 + e−x N

=N

∆2 −∆/(kB T ) e kB T 2

∆2 ex + e−x + 4 kB T 2 (ex + 1 + e−x )2 for T → 0 (12)

33

Problems in Chapter 27 (“superconductivity”)

The figure to the left shows the reduced heat capacity c = C/(N kB ) as a function of the reduced temperature t = kB T /∆ – the thin lines are the results for c(t) in the high- and low-temperature limits, (6) and (12). This characteristic behaviour of the heat capacity is called a Schottky anomaly. The number of levels involved in the anomaly for an arbitrary value of S is (2S + 1), i.e. 3 levels in the present case of S = 1. This number may be determined from a heat capacity experiment by evaluating the entropy in the high temperature limit, since S∞ = N kB ln(2S + 1) for a general value of S. A measurement of C = T (∂ S/∂T ) determines S, and

S=

T C 0

T

⇒

dT

s∞ =

S∞ = N kB

Heat capcity c

∞ c 0

t

dt = ln(2S + 1)

(13)

Entropy s

0.6

1

0.5

0.8

0.4

0.6

0.3

0.4

0.2

0.2

0.1 1

2

t

3

1

2

3

4

t

Solutions to the problems in Chapter 27 27.1 Superconducting sphere The free-energy density of a type-I superconductor is determined by the critical = 0 throughout the volume of the type-I superconductor, when field Hc . Since B surface effects are neglected (λL ≈ 0 compared to the dimensions of the sample) i = H i + 4π M = 0 B

=− 1 H M 4π i

⇒

(1)

Assuming the sample to be a thin needle along the direction of the applied magnetic ≈ 0. In this case, = −N M (along z), the demagnetization field H field H 0 z d =H and the magnetization is M = the magnetic field within the sample is H 0 i /(4π). According to eq. (2.6) in the note “Magnetic energy and domains”, the −H 0 magnetic energy density is then, 1 F = GM = − V M

sample

dr V

· δH = 1 M 0 4π

H 0 0

2 · dH = H0 H 0 0 8π

(2)

is the independent [This energy density corresponds to G in Marder because H 0 variable, see problem 27.2]. The total energy density is the sum of GM and the field-independent condensation-energy density, GS . Since the two contributions just outbalance each other at the critical field H0 = Hc , we have a measure for GS ,

G = GS + GM ,

GS = −

Hc2 8π

(3)


34

If the sample has instead the shape of a sphere, then (1) and (3) still apply, but =H + DH =H +H =H −N M , H i 0 d 0 z 0 i

D≡

Nz 1 = 4π 3

(sphere)

(4)

implying i = H

H 0 , 1−D

G = GS + GM = −

H02 Hc2 + 8π 8π(1 − D)

(5)

The internal field Hi is larger than the applied field and becomes equal the critical one, when the applied field is H0 = (1 − D)Hc = 23 Hc . This condition is not necessarily critical, since the √ global free energy of the superconducting sphere is still negative (as long as H0 < 1 − D Hc ). A detail discussion of the intermediate state of the superconductor, when (1 − D)Hc < H0 < Hc , may be found in Landau and Lifshits, Electrodynamics of continuous media (Volume 8, 2nd Edition, page 189). In the intermediate range of the applied field, local thermodynamic instabilities are going to destroy superconductivity in parts of the sphere such that the effective D is reduced sufficiently to sustain superconductivity in the remaining parts. The assumption that the superconducting domain takes the form of a prolate ellipsoid with radius a < R and constant height 2R (the largest volume for a certain value of D) is the most obvious one, but this configuration turns out to be thermodynamical unstable (the field in the normal part of the sphere is smaller than Hc near the end points of the ellipsoid). Nevertheless, domains of superconductivity are going to be present as long as H0 < Hc .

27.2 Energy of normal-superconducting interfaces

(a) To start with we consider the limit of κ → 0 or λL ξ. Then B = 0 for x > 0 and according to (27.41) the wave function, which vanishes at x = 0 and saturates at Ψ0 at large x, is

x Ψ(x) = Ψ0 ψ(x) = Ψ0 tanh √ 2ξ

(1)

The free energy density, the Ginzburg–Landau energy is, (27.27),

F=

Ψ20

L dx 0

L

2

αψ +

2 4 1 2 β Ψ0 ψ

2 + (ψ )2 2m∗

(2)

35


Here Ψ20 = −α/β is the equilibrium value at large x, and the change in energy per unit area, in comparison with the homogeneous case, is L∆F =

Ψ20

∞

2

dx α (ψ − 1) +

0

2 4 1 2 β Ψ0 (ψ

2 − 1) + (ψ )2 2m∗

(3)

Introducing Hc2 = 4πα2 /β, (27.35), and ξ 2 = −2 /(2m∗ α), (27.31), then (3) may be written L∆F =

Hc2 4π

∞ 0

dx

1 2

1 − ψ2

2

+ ξ 2 (ψ )2 =

Hc2 4π

∞ 0

dx 1 − ψ2

2

(4)

√ using ψ = (1 − ψ2 )/( 2ξ), (27.40), in the last step. Applying this relation ones more, we get √ √ √ Hc2 ∞ Hc2 2 ξ 1 Hc2 2 ξ 2 2 2 (5) dx 1 − ψ 2 ξψ = 1 − ψ dψ = L∆F = 4π 0 4π 4π 3 0 which energy per unit area is positive and in agreement with (27.52) [∆G = ∆F = 0]. since B (b) Next we consider the opposite limit of ξ λL , in which case Ψ(x) Ψ0 and the gradient ψ may be neglected. In this case the free energy density is

F = FS + FM ,

FM =

dr V

B2 4e2 A2 Ψ20 + 8π 2m∗ c2

=

dr 8πV

A2 B + 2 λL

2

(6)

where the first term is the zero-field contribution of the homogeneous superconductor, (2) when ψ = 0, and FM is the magnetic energy density in (27.27), when λ2L = m∗ c2 /(16πe2 Ψ20 ), (27.32), is inserted. The applied field is assumed to be =H in the normal metal [this conparallel to the interface, i.e. along z, and B 0 to be along the figuration leads to the largest possible energy gain]. Assuming A y direction, A = (0, A(x), 0), then B = ∇ × A = (0, 0, A (x)) for x > 0. According to (27.9), Bz = B(0) e−x/λL for x > 0, where B(0) = H0 . Hence A (x) = B(x) = H0 e−x/λL or

⇒

A(x) = −λL H0 e−x/λL (→ 0 for x → ∞)

[A(x)/λL ]2 = B 2 (x) = H02 e−2x/λL

(7)

introducing this in (6), we get (for the field energy within the superconductor) L FM =

L 0

dx

2B 2 (x) = 8π

∞ 0

dx

H02 −2x/λL H2 = 0 λL e 4π 8π

(8)

The magnetic free energy may also be calculated in a direct fashion

FM

1 = 4π

dr V

· δB = 1 H 4π

∞ H0 dx 0

L

0

−x/λL

He

dH =

H02 λL 8π L

(9)

= e−x/λL δH, and that the H-field when using δB within the superconductor is is the same as the applied field in the normal metal. In this free energy density B the independent variable. The thermodynamical potential which is at a minimum is the in the equilibrium state, is the one where the applied magnetic field H


36

independent variable. The relevant thermodynamic potential to be minimized is obtained by a Legendre transformation:

= FM

1 1 dr dr · δH H ·B B =− 4π V 4π V 1 ∞ dx 2 −λL /x H 2 λL H 2 λL − = FM − 0 H0 e =− 0 4π 0 L 4π L 8π L

GM = FM −

(10)

The magnetization in the superconductor is along the z axis and is determined =H + 4π M to be M = M (x) = −(1 − e−x/λL )H /(4π). The allowance from B 0 of the magnetic B-field to penetrate into the superconductor gives rise to the = 0 (or energy density gain determined by (10) in comparison with the case of B λL = 0). Based on eq. (2.6) in the note “Magnetic energy and domains”, this gain of magnetic energy density may be calculated in a direct fashion as

dr · δH + dr · δH M M V V 0 B= L H L 0 H dx H dx H0 = dH 1 − e−x/λL dH − 4π 4π 0 L 0 0 L 0 ∞ H 2 λ H dx H0 L = =− 0 = GM dH −e−x/λL L 4π 8π L 0 0

1 ∆FM = − V

(11)

= −N M is neglected in all the calculations, i.e. [The demagnetization field H d the sample is assumed to be a thin rod along the field, see problem 27.1]. In case the applied √ field is equal to the critical field, H0 = H√c , the interface is stable if, roughly, 2 2 ξHc2 /(12π) − λL Hc2 /(8π) < 0 or if κ > 4 2/3. This value is reduced when including interference between the two contributions, i.e. that λL becomes larger close to the surface where |Ψ| is reduced. 27.3 Diffraction effects in Josephson junctions The figure shows a weak link of normal metal (dashed lines) between two parts of a superconductor. The mag is applied in the z direcnetic field H = 0 within the superconduction. B = H in the normal tors, whereas B metal. The magnetic flux through the weak link is Φ = AB = AH, where the area is A = dx dy .

(a) With the choice of the Landau gauge, the vector potential within the normal metal is ⇒

= (0, x H, 0), A

= (0, 0, H) = B, ∇×A

=0 ∇·A

(1)

(see also problem 25.4). (b) The path integral, calculated at a constant x, between the two superconductors 1 and 2 is 2 dy = ds · A x H dy = Hdy x (2) 1

0

37


The current density in the y direction between the two superconductors, (27.68a), is independent of z (when staying within the cross-section of the normal metal):

2e j(x) = j0 sin φ2 − φ1 + c

2 1

= j sin φ − φ + 4π Hd x ds · A 0 2 1 y Φ0

(3)

The magnetic flux quantum is here defined, (25.51), Φ0 = hc/e, however, in much of the literature on superconductivity, the flux quantum is defined as the one applying for Cooper-pairs, i.e. Φ0 = hc/e∗ . The total current is d z

d x

dx

4πHdy J= dz dx j(x) = dz j0 − cos φ2 − φ1 + x Φ0 0 0 ( 4πΦ Φ0 = (dz dx )j0 cos(φ2 − φ1 ) − cos φ2 − φ1 + 4πΦ Φ0 Φ0 4πHdy

0

(4)

when introducing the total magnetic flux Φ = dx dy H. Defining J0 = dz dx j0 and using the general cosine relation: cos(u − v) − cos(u + v) = 2 sin u sin v, we finally get 2πΦ Φ0 2πΦ sin φ2 − φ1 + sin (5) J = J0 2πΦ Φ0 Φ0 (c) At zero voltage the phases of the wave functions in the superconductors 1 and 2, φ1 and φ2 , are constant in time. The result (5) clearly shows that the maximum current at a certain field and zero voltage is )

Jmax

)

Φ ) 2πΦ )) = J0 0 ))sin 2πΦ Φ0 )

(6)

Condensed Matter, Marder, Solutions

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